Questions — OCR MEI C2 (454 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
OCR MEI C2 2005 June Q3
3 Given that \(\sin \theta = \frac { \sqrt { 3 } } { 4 }\), find in surd form the possible values of \(\cos \theta\).
OCR MEI C2 2005 June Q4
4 A curve has equation \(y = x + \frac { 1 } { x }\).
Use calculus to show that the curve has a turning point at \(x = 1\).
Show also that this point is a minimum.
OCR MEI C2 2005 June Q5
5
  1. Write down the value of \(\log _ { 5 } 5\).
  2. Find \(\log _ { 3 } \left( \frac { 1 } { 9 } \right)\).
  3. Express \(\log _ { a } x + \log _ { a } \left( x ^ { 5 } \right)\) as a multiple of \(\log _ { a } x\).
OCR MEI C2 2005 June Q6
6 Sketch the graph of \(y = 2 ^ { x }\).
Solve the equation \(2 ^ { x } = 50\), giving your answer correct to 2 decimal places.
OCR MEI C2 2005 June Q7
7 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 } { x ^ { 3 } }\). The curve passes through \(( 1,4 )\).
Find the equation of the curve.
OCR MEI C2 2005 June Q8
8
  1. Solve the equation \(\cos x = 0.4\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
  2. Describe the transformation which maps the graph of \(y = \cos x\) onto the graph of \(y = \cos 2 x\).
OCR MEI C2 2005 June Q9
4 marks
9 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{faeaf2aa-ed4e-4926-b402-40c4c9aad479-3_535_790_450_630} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure} Fig. 9 shows a sketch of the graph of \(y = x ^ { 3 } - 10 x ^ { 2 } + 12 x + 72\).
  1. Write down \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Find the equation of the tangent to the curve at the point on the curve where \(x = 2\).
  3. Show that the curve crosses the \(x\)-axis at \(x = - 2\). Show also that the curve touches the \(x\)-axis at \(x = 6\).
  4. Find the area of the finite region bounded by the curve and the \(x\)-axis, shown shaded in Fig. 9 . [4]
OCR MEI C2 2005 June Q10
10 Arrowline Enterprises is considering two possible logos: \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{faeaf2aa-ed4e-4926-b402-40c4c9aad479-4_1123_1676_356_230} \captionsetup{labelformat=empty} \caption{Fig. 10.1}
\end{figure} Fig. 10.2
  1. Fig. 10.1 shows the first logo ABCD . It is symmetrical about AC . Find the length of AB and hence find the area of this logo.
  2. Fig. 10.2 shows a circle with centre O and radius 12.6 cm . ST and RT are tangents to the circle and angle SOR is 1.82 radians. The shaded region shows the second logo. Show that \(\mathrm { ST } = 16.2 \mathrm {~cm}\) to 3 significant figures.
    Find the area and perimeter of this logo.
OCR MEI C2 2005 June Q11
11 There is a flowerhead at the end of each stem of an oleander plant. The next year, each flowerhead is replaced by three stems and flowerheads, as shown in Fig. 11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{faeaf2aa-ed4e-4926-b402-40c4c9aad479-5_501_1102_431_504} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure}
  1. How many flowerheads are there in year 5?
  2. How many flowerheads are there in year \(n\) ?
  3. As shown in Fig. 11, the total number of stems in year 2 is 4, (that is, 1 old one and 3 new ones). Similarly, the total number of stems in year 3 is 13 , (that is, \(1 + 3 + 9\) ). Show that the total number of stems in year \(n\) is given by \(\frac { 3 ^ { n } - 1 } { 2 }\).
  4. Kitty's oleander has a total of 364 stems. Find
    (A) its age,
    (B) how many flowerheads it has.
  5. Abdul's oleander has over 900 flowerheads. Show that its age, \(y\) years, satisfies the inequality \(y > \frac { \log _ { 10 } 900 } { \log _ { 10 } 3 } + 1\).
    Find the smallest integer value of \(y\) for which this is true.
OCR MEI C2 2006 June Q1
1 Write down the values of \(\log _ { a } a\) and \(\log _ { a } \left( a ^ { 3 } \right)\).
OCR MEI C2 2006 June Q2
2 The first term of a geometric series is 8 . The sum to infinity of the series is 10 .
Find the common ratio.
\(3 \theta\) is an acute angle and \(\sin \theta = \frac { 1 } { 4 }\). Find the exact value of \(\tan \theta\).
OCR MEI C2 2006 June Q4
4 Find \(\int _ { 1 } ^ { 2 } \left( x ^ { 4 } - \frac { 3 } { x ^ { 2 } } + 1 \right) \mathrm { d } x\), showing your working.
OCR MEI C2 2006 June Q5
5 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 - x ^ { 2 }\). The curve passes through the point \(( 6,1 )\). Find the equation of the curve.
OCR MEI C2 2006 June Q6
6 A sequence is given by the following. $$\begin{aligned} u _ { 1 } & = 3
u _ { n + 1 } & = u _ { n } + 5 \end{aligned}$$
  1. Write down the first 4 terms of this sequence.
  2. Find the sum of the 51st to the 100th terms, inclusive, of the sequence.
OCR MEI C2 2006 June Q7
7
  1. Sketch the graph of \(y = \cos x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
    On the same axes, sketch the graph of \(y = \cos 2 x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\). Label each graph clearly.
  2. Solve the equation \(\cos 2 x = 0.5\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
OCR MEI C2 2006 June Q8
8 Given that \(y = 6 x ^ { 3 } + \sqrt { x } + 3\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
OCR MEI C2 2006 June Q9
9 Use logarithms to solve the equation \(5 ^ { 3 x } = 100\). Give your answer correct to 3 decimal places. Section B (36 marks)
OCR MEI C2 2006 June Q10
10
  1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{d697f451-9d41-4ef6-a3b0-0ebb0108932c-3_496_1029_395_516} \captionsetup{labelformat=empty} \caption{Fig. 10.1}
    \end{figure} At a certain time, ship S is 5.2 km from lighthouse L on a bearing of \(048 ^ { \circ }\). At the same time, ship T is 6.3 km from L on a bearing of \(105 ^ { \circ }\), as shown in Fig. 10.1. For these positions, calculate
    (A) the distance between ships S and T ,
    (B) the bearing of S from T .
  2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{d697f451-9d41-4ef6-a3b0-0ebb0108932c-3_444_1025_1487_520} \captionsetup{labelformat=empty} \caption{Fig. 10.2}
    \end{figure} Ship S then travels at \(24 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) anticlockwise along the arc of a circle, keeping 5.2 km from the lighthouse L, as shown in Fig. 10.2. Find, in radians, the angle \(\theta\) that the line LS has turned through in 26 minutes.
    Hence find, in degrees, the bearing of ship S from the lighthouse at this time.
OCR MEI C2 2006 June Q11
11 A cubic curve has equation \(y = x ^ { 3 } - 3 x ^ { 2 } + 1\).
  1. Use calculus to find the coordinates of the turning points on this curve. Determine the nature of these turning points.
  2. Show that the tangent to the curve at the point where \(x = - 1\) has gradient 9 . Find the coordinates of the other point, \(P\), on the curve at which the tangent has gradient 9 and find the equation of the normal to the curve at P . Show that the area of the triangle bounded by the normal at P and the \(x\) - and \(y\)-axes is 8 square units.
OCR MEI C2 2007 June Q1
1
  1. State the exact value of \(\tan 300 ^ { \circ }\).
  2. Express \(300 ^ { \circ }\) in radians, giving your answer in the form \(k \pi\), where \(k\) is a fraction in its lowest terms.
OCR MEI C2 2007 June Q2
2 Given that \(y = 6 x ^ { \frac { 3 } { 2 } }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
Show, without using a calculator, that when \(x = 36\) the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) is \(\frac { 3 } { 4 }\).
OCR MEI C2 2007 June Q3
3 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2bdf241f-4538-4227-ba00-fe843d1b3aca-2_830_1393_959_334} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Fig. 3 shows sketches of three graphs, A, B and C. The equation of graph A is \(y = \mathrm { f } ( x )\). State the equation of
  1. graph B ,
  2. graph C.
OCR MEI C2 2007 June Q4
4
  1. Find the second and third terms of the sequence defined by the following. $$\begin{aligned} t _ { n + 1 } & = 2 t _ { n } + 5
    t _ { 1 } & = 3 \end{aligned}$$
  2. Find \(\sum _ { k = 1 } ^ { 3 } k ( k + 1 )\).
OCR MEI C2 2007 June Q5
5 A sector of a circle of radius 5 cm has area \(9 \mathrm {~cm} ^ { 2 }\).
Find, in radians, the angle of the sector.
Find also the perimeter of the sector.
OCR MEI C2 2007 June Q6
6
  1. Write down the values of \(\log _ { a } 1\) and \(\log _ { a } a\), where \(a > 1\).
  2. Show that \(\log _ { a } x ^ { 10 } - 2 \log _ { a } \left( \frac { x ^ { 3 } } { 4 } \right) = 4 \log _ { a } ( 2 x )\).